The phenomenological observations of GRBs can be well described by a relativistically expanding fireball of electrons, photons, and protons [1]}, [2]}, [3]}. Ultra-high energy neutrinos can be emitted during the fireball expansion [4]}, [5]}, [6]}, [7]}. From the viewpoint of the fireball model, the accelerated protons can scatter with the intense \(\gamma \) -ray background within the GRB fireball, and generate pions:
\(p+\gamma \rightarrow \Delta ^+\rightarrow n+\pi ^+.\) 
